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We may now examine the implications of the axioms in the context of the properties of indifference curves. In this context we also refer to a few additional axioms. We start with the implications of the axiom of non-satiation. Recall that a consumption bundle X is preferred to Y if it contains more of at least one good and no less of the other, i.e., if X > Y.
The non-satiation assumption has two important consequences for the nature of indifferent sets. In Fig. 4.16 X = (x’1, x’2) is a consumption bundle. The axiom of non-satiation implies that in the area B (including the boundaries, except for X itself) must be preferred to X, and all points in the area W (again including the boundaries except for X) must be inferior to X.
This axiom implies that points in the indifference set for X (if there are any besides X) must lie in areas A and C. In other words, if a consumer moves among bundles in the indifference set, he can only do this by substituting or trading off the goods — giving more of one good must require taking away some of the other good so that he can stay within the indifferent set.
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An indifference set can never be wider than a single point. This means that an indifference set may be a single point, an unconnected set of points or a curve. If some bundles indifferent to X (which is contained in an indifference set) lie in areas B and W, the non-satiation assumption is violated.
The so-called lexicographic ordering satisfies assumptions 1 to 4, but each of its indifference sets consist of only one point (Fig. 4.17). However, since we use indifference curves to show a consumer’s choice problem, from the point of view of solving optimisation problems continuity is a very important property. Hence, we use the axiom of continuity.
Axiom of Continuity:
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The graph of an indifference map is a continuous surface. This means that any indifference curve has no gaps or breaks at any point. In terms of the consumer’s choice behaviour, it implies. Given two goods in his consumption bundle, we can always find an increase in the other good which will exactly compensate him, i.e., leave him with a consumption bundle indifferent to the first. This is possible only if the indifference surface is everywhere continuous.
Deriving the Properties of the Indifference Map of a Consumer:
All the above axioms are related to a commodity space and to a consumer’s preferences. Now in order to relate directly the consumer’s preferences to the commodity space of Fig. 4.18, we have to introduce a further axiom.
Axiom of Dominance:
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In Fig. 4.18 since bundle X has more of both commodities, X dominates Y. If X dominates Y the consumer will prefer X;
Thus X > Y implies that X is preferred to Y.
In short, the consumer always prefers more of both commodities to less. Even if X has more of one commodity and the same amount of the other, X will dominate Y (weak dominance) as Fig. 4.19 shows. Here X dominates Y, as does Z. In fact, any point in the shaded area with Y as origin dominates Y. In the same way, any point in the lower (south-west) quadrant such as K is inferior to Y since Y dominates all points in it.
If the axiom of dominance is modified to some extent, any indifference curve will be thick, as illustrated in Fig. 4.20. Here we get a range of a thick region (called band). It is not a boundary line separating preferred bundles from non-preferred ones. The points lying inside the band are indifferent to one another.
Such a situation is encountered when the difference in utility between two points Y and X is so small that even a rational consumer cannot perceive it, i.e., feel the difference. This is why we get an indifference region rather than an indifference curve.
Axiom of Non-Satiation (Monotonicity):
The axiom of dominance is also known as the axiom of non-satiation or the axiom of monotonicity. The axiom holds only for economic goods. For economic ‘bads’ such as garbage or polluted air we are likely to get the opposite dominance — less of both or at least one is preferred to more of both or at least one.
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In Fig. 4.20, compared to Y each zone of ignorance contains less of one commodity and more of the other. Point W, for example, has more of x2 and less of x1. Point S has more of x1 and less of x2. The axiom of dominance does not enable us to say anything about this point.
In Fig. 4.21 we pass a line through the north-west zone of ignorance, but also through both the zones which are inferior and superior to X, respectively. The axiom of dominance suggest that all points on the line segment YZ are preferred to X, simply because YZ lies in the superior quadrant with X as origin.
Similarly all points on 0W are inferior to X. But a point like Y will surely be preferred to W, since Flies north-west of W: it contains more of both x1 and x2.
This means that, in-between W and Y there is a point which indicates a certain change of preferences up to W point X preferred, whereas from Y onwards each point on the ray 0Z is preferred 10 X. This implies that there must be a point where this switch takes place and this point must lie on WY. As long as this preference ordering changes smoothly, there will surely exist a point, such as M, which is indifferent to X.
If we repeat this exercise with M as the reference point, there is likely to exist a point like N which is indifferent to M. Then with N as reference point, we can establish P which is indifferent to N and so on. The continuous line (the ‘locus’ joining P and M) and X with similar points in the south-east quadrant is obviously an indifference curve.
The locus of all commodity combinations from which the consumer derives the same level of satisfaction forms such a curve.
An indifference map, is collection of indifference curves corresponding to different levels of satisfaction. The indifference curve is a boundary line: to the right of the line we have a set of points which are preferred to the set up points to the left of the line. On the line itself, all points are indifferent to one another. And all the points below the line are inferior to all the points on the line. See Fig. 4.22.
So far we have established only that the line slopes downward from left to right. It could have any of the shapes shown in Fig. 4.22 (or even any combination of these shapes).
A Counter Example:
In case of lexicographic orderings, an indifference curve does not exists in the true sense. The lexicographic ordering is complete, transitive, strongly monotone and strictly convex. Yet no utility function exists that represents this preference ordering. Suppose the consumer prefers any bundle with more x1, regardless of the amount of x2 in the bundle.
At the same time, if two bundles contain an equal amount of x1 we can assume that the consumer prefers the bundle with more x2. This kind of ordering is shown in Fig. 4.23.
To the right of X, all bundles contain more x1; hence all points to the right of x1, regardless of which quadrant they are in, are preferred to X. Similarly, all points to the left of X are inferior to X. For bundles with a given amount of x1 — that is, bundles lying on the vertical line through X — those to the north are preferred to those to the south.
Now point Y is not indifferent to X but is inferior to X because it lies to the left of X. Points like Z and W arc superior and inferior, respectively. In short, there are no points, other than X itself, which are indifferent to X. There is thus no indifference curve.
Thus to be able to derive an indifference curve we have to rule out the possibility of lexicographic orderings (which virtually amounts to ignoring addicts, whether it is cigarette, alcohol or chewing gum). No utility function represents this preference ordering.
With this preference ordering, no two distinct bundles are indifferent; indifferent sets are singletons. The assumption that is needed to ensure the existence of a utility function is that the preference relation be continuous. In order to ensure that we have indifference curves like those in Fig. 4.21 we have to set another axiom.
The Axiom of Continuity of Preference:
There exists a set of points on a boundary dividing the commodity space into less preferred (inferior) and more preferred (superior) areas such that these points are indifferent to one another. Fig. 4.24, where the indifference curve is smooth and continuous.
Convex Preferences:
Fig 4.22 shows three possible shapes for an indifference curve. Of these only curve IC] is meaningful because it is convex to the origin. In order to ensure this shape of an indifference curve we have to make a further axiom.
The Axiom of Convexity of Preferences:
To restrict ourselves to a curve like IC1 we have to draw a distinction between ‘general’ convexity and ‘strict’ convexity. In Fig. 4.22 while IC3 is also convex, IC1 is strictly convex. In Fig. 4.25 for a movement along IC from left to right a certain amount of x1 is required to compensate the consumer for the loss of x2.
But for a movement along IC from Y to Z a larger amount of x2 is required by the consumer to compensate for the loss of x1 than for a movement from X to Y. The magnitude |∆x2/∆x1|, called MRS, becomes smaller.
In other words, MRS diminishes along the same indifference curve. The reason is that as the consumer has less and less of x2 he will require successively larger and larger amounts of x1 to compensate him for the loss of x2. The less the consumer’s possession of x2, the more highly valued its last unit will be to him. This is supposed to be a common feature of consumer preferences.
The Axiom of Strict Convexity:
If and only if indifference curves are strictly convex, they are smooth. The axiom of strict convexity suggest that, given any consumption bundle Y, its better set a strictly convex. In Fig. 4.27 the better set for point Y is the set of points on the indifference curve IC and in the shaded area (such as point B) and this is drawn as strictly convex.
‘Convexity’ does not rule out the possibility of indifference curves that are completely linear (i.e., straight lines) or indifference curves that are ‘piecewise linear’ (i.e., have linear segments), like the ones shown in Fig. 4.26. To ensure smooth convexity of an indifference curve, we have to make the assumption of ‘strict convexity’.
In Fig. 4.27 the line XY joints the two ‘end points’, X and Y. A point intermediate between X and Y is given by W = (1 — α) X + αY, where 0 < α < 1. W is then preferred to X and Y. Since W is a weighted average of X and Y, the convexity assumption is stated in terms of ‘preferences for averages (means) over extremes.’ If W lies to the right of indifference curve, the curve is strictly convex.
The strict convexity axiom can be expressed as:
(1 – α)X + αY > X(or Y)
an average bundle > any extreme bundle.
The axiom of strict convexity implies that any mixture along the line XWY will be preferred to Z and Y. Thus, the consumer always prefers a mixture of two consumption bundles which are indifferent to each other, to either one of those bundles. This preference for mixtures is a commonly observed aspect of consumer behaviour.
If, however, while constructing the segment XY we find that IV lies on the indifference curve, the curve is simply convex, not strictly so. In other words, the term ‘convex’ covers both the strict convex case and the case where W lies on the indifference curve.
In Fig. 4.27 W will lie on the indifference curve if the curve is linear. This is a case of weak convexity.
The axiom of weak convexity suggests that the better set is convex but not strictly so. This means that we allow the possibility of line segments in the indifference curves as Fig. 4.28 shows. The better sets for points X, Y and Z, respectively are convex but none is strictly convex.
Linearity in the indifference curve over some range implies that within this range, MRS remains constant. This means that successive equal reductions in the amount of x2 are compensated by successive equal increases in the amount of x1. Alternatively, a mixture of two indifferent bundles is indifferent to the two extreme bundles, rather than preferred to them.
Another Implication of Strict Convexity:
Another implication of strict convexity is that the indifference curves cannot cut the axes. If they did, the axes would become extensions of the indifference curves. But, since the axes are linear, this is inconsistent with strict convexity, although it is consistent with (weak) convexity. Hence, strict convexity rules out the possibility of indifference curves cutting the axes.
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